(а)In (3x + 5)limx0In(7x + 3) +1(b)1 xlim (sin(x)) ·(c)lim(x - Vx2 + 1)

Answered: (a) In (3x + 5) lim x0o In(7x + 3) + 1

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.5: Rational Functions

Problem 8E

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Question

(а)
In (3x + 5)
lim
x0In(7x + 3) +1
(b)
1 x
lim (sin(x)) ·
(c)
lim(x - Vx2 + 1)

Expert Solution

Step 1: Given that:

“Since you have asked multiple questions, we will solve the first question for you. If you want any specific question to be solved then please specify the question number or post only that question.”

Step 2: Part (a)

We have,

$\lim_{x \to \infty} \frac{\ln (3 x + 5)}{\ln (7 x + 3) + 1}$

Since on the given limit, we have the Indeterminate form of the type $\frac{\infty}{\infty}$ so we can apply L'Hopital's Rule:

$\begin{array}{rcl} \lim_{x \to \infty} \frac{\ln (3 x + 5)}{\ln (7 x + 3) + 1} & = & \lim_{x \to \infty} \frac{\frac{d}{d x} \{\ln (3 x + 5)\}}{\frac{d}{d x} \{\ln (7 x + 3) + 1\}} \\ = & \lim_{x \to \infty} \frac{\frac{3}{3 x + 5}}{\frac{7}{7 x + 3}} \\ = & \lim_{x \to \infty} \frac{3 (7 x + 3)}{7 (3 x + 5)} \\ = & \lim_{x \to \infty} \frac{3 x (7 + \frac{3}{x})}{7 x (3 + \frac{5}{x})} \\ = & \lim_{x \to \infty} \frac{3 (7 + \frac{3}{x})}{7 (3 + \frac{5}{x})} \\ = & \frac{3 (7 + \lim_{x \to \infty} \frac{3}{x})}{7 (3 + \lim_{x \to \infty} \frac{5}{x})} \\ = & \frac{3 (7 + 0)}{7 (3 + 0)} . . . . (S i n c e \frac{1}{\infty} = 0) \\ = & \frac{3 \times 7}{7 \times 3} \\ = & 1 \end{array}$

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